Note that the non-spiked part \( \sum _{j=M+1}^{p}f\left(\lambda _{j} \right) \)  cannot use the conclusion of [1]} or [2]} directly, because the off-diagonal block \( {\bf S}_{12} \)  and \( {\bf S}_{21} \)  are not 0, while the linear spectral statistics generated by \( {\bf S}_{22} \)  can use the conclusion of [1]} or [2]} directly, so in the following lemma we calculate the difference between \( \sum _{j=M+1}^{p}f\left( \lambda _{j}\right)\)  and \(\sum _{j=1}^{p-M}f\left(\widetilde{{\bf \lambda }_{j}} \right) \) .
